Structural Uncertainty and the Value of Statistical Life in the Economics of Catastrophic Climate Change

Using climate change as a prototype motivating example, this paper analyzes the implications of structural uncertainty for the economics of low-probability high-impact catastrophes. The paper shows that having an uncertain multiplicative parameter, which scales or amplifies exogenous shocks and is updated by Bayesian learning, induces a critical "tail fattening" of posterior-predictive distributions. These fattened tails can have strong implications for situations (like climate change) where a catastrophe is theoretically possible because prior knowledge cannot place sufficiently narrow bounds on overall damages. The essence of the problem is the difficulty of learning extreme-impact tail behavior from finite data alone. At least potentially, the influence on cost-benefit analysis of fat-tailed uncertainty about the scale of damages -- coupled with a high value of statistical life -- can outweigh the influence of discounting or anything else.

Coherent Asset Allocation and Diversification in the Presence of Stress Events

We propose a method to integrate frequentist and subjective probabilities in order to obtain a coherent asset allocation in the presence of stress events. Our working assumption is that in normal market asset returns are sufficiently regular for frequentist statistical techniques to identify their joint distribution, once the outliers have been removed from the data set. We also argue, however, that the exceptional events facing the portfolio manager at any point in time are specific to the each individual crisis, and that past regularities cannot be relied upon. We therefore deal with exceptional returns by eliciting subjective probabilities, and by employing the Bayesian net technology to ensure logical consistency. The portfolio allocation is then obtained by utility maximization over the combined (normal plus exceptional) distribution of returns. We show the procedure in detail in a stylized case.Stress tests, asset allocation, Bayesian Networks

Real Time Detection of Structural Breaks in GARCH Models

A sequential Monte Carlo method for estimating GARCH models subject to an unknown number of structural breaks is proposed. Particle filtering techniques allow for fast and efficient updates of posterior quantities and forecasts in real-time. The method conveniently deals with the path dependence problem that arises in these type of models. The performance of the method is shown to work well using simulated data. Applied to daily NASDAQ returns, the evidence favors a partial structural break specification in which only the intercept of the conditional variance equation has breaks compared to the full structural break specification in which all parameters are subject to change. The empirical application underscores the importance of model assumptions when investigating breaks. A model with normal return innovations result in strong evidence of breaks; while more flexible return distributions such as t-innovations or a GARCH-jump mixture model still favor breaks but indicate much more uncertainty regarding the time and impact of them.particle filter, GARCH model, change point, sequential Monte Carlo

Heavy-tailed distributions in VaR calculations

The essence of the Value-at-Risk (VaR) and Expected Shortfall (ES) computations is estimation of low quantiles in the portfolio return distributions. Hence, the performance of market risk measurement methods depends on the quality of distributional assumptions on the underlying risk factors. This chapter is intended as a guide to heavy-tailed models for VaR-type calculations. We first describe stable laws and their lighter-tailed generalizations, the so-called truncated and tempered stable distributions. Next we study the class of generalized hyperbolic laws, which – like tempered stable distributions – can be classified somewhere between infinite variance stable laws and the Gaussian distribution. Then we discuss copulas, which enable us to construct a multivariate distribution function from the marginal (possibly different) distribution functions of n individual asset returns in a way that takes their dependence structure into account. This dependence structure may be no longer measured by correlation, but by other adequate functions like rank correlation, comonotonicity or tail dependence. Finally, we provide numerical examples.Heavy-tailed distribution; Stable distribution; Tempered stable distribution; Generalized hyperbolic distribution; Parameter estimation; Value-at-Risk (VaR); Expected Shortfall (ES); Copula; Filtered historical simulation (FHS);